Ian Simplification

Percentage Accurate: 6.9% → 8.4%

Time: 42.0s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 8.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, -2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow
   (cbrt
    (cbrt
     (pow
      (fma PI 0.5 (* (- (* PI 0.5) (acos (sqrt (+ 0.5 (* x -0.5))))) -2.0))
      2.0)))
   3.0)
  (cbrt (fma PI 0.5 (* -2.0 (asin (sqrt (- 0.5 (* 0.5 x)))))))))

C

double code(double x) {
	return pow(cbrt(cbrt(pow(fma(((double) M_PI), 0.5, (((((double) M_PI) * 0.5) - acos(sqrt((0.5 + (x * -0.5))))) * -2.0)), 2.0))), 3.0) * cbrt(fma(((double) M_PI), 0.5, (-2.0 * asin(sqrt((0.5 - (0.5 * x)))))));
}

Julia

function code(x)
	return Float64((cbrt(cbrt((fma(pi, 0.5, Float64(Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 + Float64(x * -0.5))))) * -2.0)) ^ 2.0))) ^ 3.0) * cbrt(fma(pi, 0.5, Float64(-2.0 * asin(sqrt(Float64(0.5 - Float64(0.5 * x))))))))
end

Wolfram

code[x_] := N[(N[Power[N[Power[N[Power[N[Power[N[(Pi * 0.5 + N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[(Pi * 0.5 + N[(-2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 9.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right) \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}} \]

   2. pow3 9.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)}^{3}} \]

4. Applied egg-rr 9.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}\right)}^{3}} \]
5. Step-by-step derivation

   1. add-cube-cbrt 9.2%

      \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}}\right)}}^{3} \]

   2. unpow-prod-down 9.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}}\right)}^{3}} \]

6. Applied egg-rr 9.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)}} \]
7. Step-by-step derivation

   1. asin-acos 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   2. sub-neg 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5 \cdot x\right)}}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   3. *-commutative 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 + \left(-\color{blue}{x \cdot 0.5}\right)}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   4. sub-neg 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - x \cdot 0.5}}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   5. div-inv 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   6. metadata-eval 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   7. sub-neg 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   8. *-commutative 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \left(-\color{blue}{0.5 \cdot x}\right)}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   9. sub-neg 10.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   10. sub-neg 10.6%

       \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5 \cdot x\right)}}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   11. *-commutative 10.6%

       \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \left(-\color{blue}{x \cdot 0.5}\right)}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   12. distribute-rgt-neg-in 10.6%

       \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

   13. metadata-eval 10.6%

       \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

8. Applied egg-rr 10.6%

   \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)} \]

9. Final simplification 10.6%

   \[\leadsto {\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot -2\right)\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, -2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]
10. Add Preprocessing

Alternative 2: 8.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (cbrt (+ (* PI 0.5) (* -2.0 (- (* PI 0.5) (acos (sqrt (- 0.5 (* 0.5 x))))))))
  3.0))

C

double code(double x) {
	return pow(cbrt(((((double) M_PI) * 0.5) + (-2.0 * ((((double) M_PI) * 0.5) - acos(sqrt((0.5 - (0.5 * x)))))))), 3.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.cbrt(((Math.PI * 0.5) + (-2.0 * ((Math.PI * 0.5) - Math.acos(Math.sqrt((0.5 - (0.5 * x)))))))), 3.0);
}

Julia

function code(x)
	return cbrt(Float64(Float64(pi * 0.5) + Float64(-2.0 * Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 - Float64(0.5 * x)))))))) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[(Pi * 0.5), $MachinePrecision] + N[(-2.0 * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 9.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right) \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}} \]

   2. pow3 9.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)}^{3}} \]

4. Applied egg-rr 9.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}\right)}^{3}} \]
5. Step-by-step derivation

   1. asin-acos 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \cdot -2\right)}\right)}^{3} \]

   2. div-inv 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot -2\right)}\right)}^{3} \]

   3. metadata-eval 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot -2\right)}\right)}^{3} \]

   4. *-commutative 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)\right) \cdot -2\right)}\right)}^{3} \]

   5. cancel-sign-sub-inv 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \cdot -2\right)}\right)}^{3} \]

   6. metadata-eval 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \cdot -2\right)}\right)}^{3} \]

   7. *-commutative 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right) \cdot -2\right)}\right)}^{3} \]

   8. +-commutative 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{x \cdot -0.5 + 0.5}}\right)\right) \cdot -2\right)}\right)}^{3} \]

   9. fma-define 10.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, -0.5, 0.5\right)}}\right)\right) \cdot -2\right)}\right)}^{3} \]

6. Applied egg-rr 10.5%

   \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)} \cdot -2\right)}\right)}^{3} \]

7. Taylor expanded in x around inf 10.5%

   \[\leadsto {\left(\sqrt[3]{\color{blue}{-2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) + 0.5 \cdot \pi}}\right)}^{3} \]

8. Final simplification 10.5%

   \[\leadsto {\left(\sqrt[3]{\pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}\right)}^{3} \]
9. Add Preprocessing

Alternative 3: 8.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) - \frac{\pi}{2}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ PI 2.0) (* 2.0 (- (acos (sqrt (+ 0.5 (* x -0.5)))) (/ PI 2.0)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (acos(sqrt((0.5 + (x * -0.5)))) - (((double) M_PI) / 2.0)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) + (2.0 * (Math.acos(Math.sqrt((0.5 + (x * -0.5)))) - (Math.PI / 2.0)));
}

Python

def code(x):
	return (math.pi / 2.0) + (2.0 * (math.acos(math.sqrt((0.5 + (x * -0.5)))) - (math.pi / 2.0)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(sqrt(Float64(0.5 + Float64(x * -0.5)))) - Float64(pi / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 + (x * -0.5)))) - (pi / 2.0)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] + N[(2.0 * N[(N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. asin-acos 10.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]

   2. add-cube-cbrt 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   3. associate-/l* 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \frac{\sqrt[3]{\pi}}{2}} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   4. fma-neg 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]

   5. pow2 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   6. div-sub 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]

   7. metadata-eval 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]

   8. div-inv 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]

   9. metadata-eval 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

4. Applied egg-rr 8.3%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]
5. Step-by-step derivation

   1. fma-neg 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]

   2. associate-*r/ 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\frac{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   3. unpow2 8.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)} \cdot \sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   4. rem-3cbrt-lft 10.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   5. sub-neg 10.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right)\right) \]

   6. distribute-rgt-neg-in 10.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)\right) \]

   7. metadata-eval 10.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)\right) \]

6. Simplified 10.5%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \]

7. Final simplification 10.5%

   \[\leadsto \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) - \frac{\pi}{2}\right) \]
8. Add Preprocessing

Alternative 4: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing

3. Final simplification 9.2%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
4. Add Preprocessing

Alternative 5: 3.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* PI 0.5) (* 2.0 (asin (sqrt 0.5)))))

C

double code(double x) {
	return (((double) M_PI) * 0.5) + (2.0 * asin(sqrt(0.5)));
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) + (2.0 * Math.asin(Math.sqrt(0.5)));
}

Python

def code(x):
	return (math.pi * 0.5) + (2.0 * math.asin(math.sqrt(0.5)))

Julia

function code(x)
	return Float64(Float64(pi * 0.5) + Float64(2.0 * asin(sqrt(0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi * 0.5) + (2.0 * asin(sqrt(0.5)));
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] + N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 9.2%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]

   2. +-commutative 9.2%

      \[\leadsto \color{blue}{\left(-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) + \frac{\pi}{2}} \]

   3. add-cube-cbrt 8.5%

      \[\leadsto \left(-\color{blue}{\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right) \cdot \sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}}\right) + \frac{\pi}{2} \]

   4. distribute-rgt-neg-in 8.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right) \cdot \left(-\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)} + \frac{\pi}{2} \]

   5. fma-define 8.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}, -\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}, \frac{\pi}{2}\right)} \]

4. Applied egg-rr 8.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}^{2}, -\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}^{2}, \color{blue}{\sqrt{-\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}} \cdot \sqrt{-\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}}, \pi \cdot 0.5\right) \]

   2. sqrt-unprod 3.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}^{2}, \color{blue}{\sqrt{\left(-\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right) \cdot \left(-\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}}, \pi \cdot 0.5\right) \]

   3. sqr-neg 3.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}^{2}, \sqrt{\color{blue}{\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \cdot \sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}}, \pi \cdot 0.5\right) \]

   4. sqrt-prod 3.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}^{2}, \color{blue}{\sqrt{\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}} \cdot \sqrt{\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}}, \pi \cdot 0.5\right) \]

   5. add-sqr-sqrt 3.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}^{2}, \color{blue}{\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}, \pi \cdot 0.5\right) \]

   6. fma-define 3.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)}^{2} \cdot \sqrt[3]{2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} + \pi \cdot 0.5} \]

6. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) + \pi \cdot 0.5} \]

7. Taylor expanded in x around 0 3.6%

   \[\leadsto 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} + \pi \cdot 0.5 \]

8. Final simplification 3.6%

   \[\leadsto \pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
9. Add Preprocessing

Alternative 6: 4.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 4.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} \]

4. Final simplification 4.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
5. Add Preprocessing

Developer target: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2024062 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))